if \(H_0(t)\)  satisfies the condition (REF ).
Here the phase of \(|n(t) {\rangle }\)  has been chosen and fixed arbitrarily.
The first term of the exponent \(\xi _n(t)\)  is the conventional dynamical
phase while the second term is the geometric phase [1]}. Let \(H_0(t)\)  depend on time through a set of parameters, collectively denoted as \({\lambda }(t)\) , namely \(H_0(t) = H_0({\lambda }(t))\) , where \({\lambda }(t) \in \mathbb {R}^N\) , \(N\)  being the number
of independent parameters. Correspondingly \(|n(t) {\rangle }\)  is also
written as \(|n({\lambda }(t)){\rangle }\) . The geometric phase is independent of the speed of time-evolution but only depends on the path in the parameter space so far as the time-evolution is adiabatic;
\(\int _0^t {\langle }n(s) |\partial _s n(s) {\rangle }ds =\int _{\lambda (0)}^{{\lambda }(t)} {\langle }n(\lambda )| \partial _{\mu } n(\lambda ) {\rangle }d\lambda ^{\mu },\) 
